A062298 -id:A062298 - cp's OEIS frontend

A254218 T(n,k) = number of length n 1..(k+2) arrays with no leading or trailing partial sum equal to a prime and no consecutive values equal.

1, 2, 0, 2, 0, 0, 3, 0, 1, 0, 3, 2, 3, 2, 1, 4, 2, 11, 4, 2, 0, 5, 8, 12, 22, 8, 2, 0, 6, 12, 32, 24, 56, 6, 2, 0, 6, 18, 48, 96, 70, 136, 15, 2, 0, 7, 18, 86, 168, 373, 192, 383, 18, 5, 0, 7, 28, 98, 388, 766, 1472, 633, 1070, 45, 4, 0, 8, 28, 172, 490, 2056, 3720, 6490, 2484, 3897

Offset: 1

R. H. Hardin, Jan 26 2015

Table starts
.1.2...2.....3.....3......4.......5........6........6.........7.........7
.0.0...0.....2.....2......8......12.......18.......18........28........28
.0.1...3....11....12.....32......48.......86.......98.......172.......183
.0.2...4....22....24.....96.....168......388......490......1024......1168
.1.2...8....56....70....373.....766.....2056.....2803......6705......8187
.0.2...6...136...192...1472....3720....11182....16698.....44652.....58174
.0.2..15...383...633...6490...18214....60168....97089....296955....420163
.0.2..18..1070..2484..28190...81428...316982...574274...2056696...3150280
.0.5..45..3897.10554.109811..362910..1788533..3605385..14593061..23955140
.0.4.118.13372.35054.428042.1828848.10469104.22736838.103347086.183929058

			Some solutions for n=4 k=4
..4....4....4....1....1....1....1....6....6....6....1....6....4....4....4....6
..6....2....2....3....5....5....5....3....3....4....5....2....2....5....6....4
..4....4....3....2....6....3....4....5....5....2....3....4....6....3....2....2
..6....6....1....4....4....6....6....1....4....4....1....6....4....6....4....6

R. H. Hardin, Table of n, a(n) for n = 1..265

Row 1 is A062298(n+2).

Column k=1 gives A254211.

A254539 T(n,k)=Number of length n 1..(k+2) arrays with no leading partial sum equal to a prime.

Original entry on oeis.org

1, 2, 1, 2, 3, 1, 3, 5, 8, 2, 3, 10, 15, 20, 4, 4, 11, 40, 45, 50, 6, 5, 20, 49, 160, 135, 126, 11, 6, 28, 105, 222, 670, 421, 329, 20, 6, 37, 163, 576, 1087, 3001, 1466, 956, 33, 7, 41, 253, 1026, 3383, 5604, 13503, 5403, 2897, 62, 7, 54, 307, 1849, 6814, 20393, 29038, 60408

Offset: 1

R. H. Hardin, Feb 01 2015

Table starts
..1....2.....2.......3.......3........4........5.........6.........6..........7
..1....3.....5......10......11.......20.......28........37........41.........54
..1....8....15......40......49......105......163.......253.......307........466
..2...20....45.....160.....222......576.....1026......1849......2461.......4195
..4...50...135.....670....1087.....3383.....6814.....13843.....20012......37643
..6..126...421....3001....5604....20393....45472....102595....161277.....338402
.11..329..1466...13503...29038...121774...297210....758766...1313695....3093457
.20..956..5403...60408..150268...709169..1936867...5719495..10916298...28507728
.33.2897.19417..270370..764508..4121638.12941917..43758333..91142820..262001403
.62.8341.69205.1192385.3857845.24622476.88456127.333905794.755234611.2410105286

			Some solutions for n=4 k=4
..4....4....6....6....6....4....4....6....1....4....6....1....4....1....6....6
..4....6....6....3....2....6....2....6....5....5....6....5....6....3....3....2
..1....5....4....5....1....2....2....2....2....1....2....6....6....5....1....4
..1....5....4....2....6....2....1....1....1....2....2....6....4....3....5....4

R. H. Hardin, Table of n, a(n) for n = 1..9999

Row 1 is A062298(n+2)

A255716 T(n,k)=Number of length n 1..(k+2) arrays with no leading partial sum equal to a prime and no consecutive values equal.

Original entry on oeis.org

1, 2, 1, 2, 2, 1, 3, 4, 4, 1, 3, 8, 10, 6, 1, 4, 9, 26, 21, 8, 1, 5, 17, 33, 81, 47, 14, 1, 6, 24, 75, 120, 261, 102, 19, 1, 6, 32, 121, 342, 480, 936, 256, 33, 1, 7, 36, 191, 653, 1707, 2079, 3435, 754, 69, 1, 7, 48, 242, 1221, 3764, 8814, 9044, 12406, 2169, 162, 1, 8, 52, 374

Offset: 1

R. H. Hardin, Mar 03 2015

Table starts
.1...2....2......3......3.......4........5.........6.........6.........7
.1...2....4......8......9......17.......24........32........36........48
.1...4...10.....26.....33......75......121.......191.......242.......374
.1...6...21.....81....120.....342......653......1221......1729......3037
.1...8...47....261....480....1707.....3764......8072.....12646.....24669
.1..14..102....936...2079....8814....22155.....53090.....91493....200228
.1..19..256...3435...9044...45735...127375....346499....667623...1659159
.1..33..754..12406..39604..229317...721134...2312940...4999570..13912571
.1..69.2169..45637.170835.1129024..4201419..15787318..37773069.116235183
.1.162.5999.165303.720348.5744720.25338485.107642427.282611786.969235198

			Some solutions for n=4 k=4
..4....4....4....6....4....1....1....6....1....1....1....1....6....1....4....1
..5....5....6....2....6....3....5....3....3....3....3....5....3....3....6....3
..3....6....4....6....4....4....2....1....2....5....6....6....1....2....2....6
..6....3....2....4....6....1....4....2....6....1....5....4....5....3....6....2

R. H. Hardin, Table of n, a(n) for n = 1..9999

Row 1 is A062298(n+2)

A239968 a(n) = 0 unless n is a nonprime A018252(k) then a(n) = k.

Original entry on oeis.org

1, 0, 0, 2, 0, 3, 0, 4, 5, 6, 0, 7, 0, 8, 9, 10, 0, 11, 0, 12, 13, 14, 0, 15, 16, 17, 18, 19, 0, 20, 0, 21, 22, 23, 24, 25, 0, 26, 27, 28, 0, 29, 0, 30, 31, 32, 0, 33, 34, 35, 36, 37, 0, 38, 39, 40, 41, 42, 0, 43, 0, 44, 45, 46, 47, 48, 0, 49, 50, 51, 0, 52

Offset: 1

Reinhard Zumkeller, Mar 30 2014

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A005171, A010051, A018252, A057427, A066246, A049084, A026233, A062298.

import Data.List (unfoldr, genericIndex)
a239968 n = genericIndex a239968_list (n - 1)
a239968_list = unfoldr c (1, 1, a018252_list) where
   c (i, z, xs'@(x:xs)) | i == x = Just (z, (i + 1, z + 1, xs))
                        | i /= x = Just (0, (i + 1, z, xs'))

Module[{k = 0}, Array[If[!PrimeQ[#], ++k, 0] &, 100]] (* Paolo Xausa, Jul 31 2025 *)

a(n) = A066246(n) - A010051(n) + 1.
a(n) = A026233(n) - A049084(n);
A057427(a(n)) = A005171(n).
a(n) = A062298(n)*A005171(n). - Ridouane Oudra, Jul 29 2025

A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 4, 5, 6, 6, 6, 7, 7, 8, 9, 10, 10, 11, 11, 12, 12, 13, 13, 14, 14, 14, 15, 16, 17, 18, 18, 19, 20, 21, 21, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 30, 31, 32, 33, 34, 34, 35, 35, 36, 37, 37, 38, 39, 39, 40, 41, 42

Offset: 1

Gus Wiseman, Jul 31 2022

nonn

			The a(30) = 14 numbers: 1, 6, 10, 12, 14, 15, 18, 20, 21, 22, 24, 26, 28, 30.

The complement is counted by A025528, with 1's A065515.
For primes instead of prime-powers we have A062298, with 1's A065855.
The version treating 1 as a prime-power is A085970.
One more than the partial sums of A143731.
A000688 counts factorizations into prime-powers.
A001222 counts prime-power divisors.
A246655 lists the prime-powers (A000961 includes 1), towers A164336.
Cf. A000720, A023894, A036234, A050361, A054685, A069513, A355743, A356064, A356065, A356066.

Table[Length[Select[Range[n],!PrimePowerQ[#]&]],{n,100}]

a(n) = A085970(n) + 1.

A065134 Remainder when n is divided by the number of primes not exceeding n.

Original entry on oeis.org

0, 1, 0, 2, 0, 3, 0, 1, 2, 1, 2, 1, 2, 3, 4, 3, 4, 3, 4, 5, 6, 5, 6, 7, 8, 0, 1, 9, 0, 9, 10, 0, 1, 2, 3, 1, 2, 3, 4, 2, 3, 1, 2, 3, 4, 2, 3, 4, 5, 6, 7, 5, 6, 7, 8, 9, 10, 8, 9, 7, 8, 9, 10, 11, 12, 10, 11, 12, 13, 11, 12, 10, 11, 12, 13, 14, 15, 13, 14, 15, 16, 14, 15, 16, 17, 18, 19, 17, 18, 19

Offset: 2

Labos Elemer, Oct 15 2001

nonn

Also remainder when the number of nonprimes is divided by the number of primes (not exceeding n).

			n = 2: Pi[2] = 1,Mod[1,1] = 0, the first term = a(2) = 0; n = 100: Pi[100] = 25, Mod[100,25] = 0 = a(100); n = 20: Pi[20] = 8, Mod[20,8] = 4 = a(20).

Harry J. Smith, Table of n, a(n) for n = 2..1000

Cf. A000720, A057809, A057810, A000040, A065134, A004648, A062298, A065858-A065864, A065133.

Table[Last@ QuotientRemainder[n, PrimePi[n]], {n, 2, 91}] (* Michael De Vlieger, Jul 04 2016 *)

{ for (n=2, 1000, write("b065134.txt", n, " ", n%primepi(n)) ) } \\ Harry J. Smith, Oct 11 2009

a(n) = n (mod pi(n)).

Term a(1) removed so OFFSET changed from 1,5 to 2,4 by Harry J. Smith, Oct 11 2009

Since OFFSET is 2,4; Term a(1) removed and a(91) added by Harry J. Smith, Oct 11 2009

A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 9, 7, 6, 8, 12, 11, 15, 23, 81, 18, 10, 17, 30, 13, 162, 27, 36, 19, 24, 16, 25, 38, 46, 37, 45, 31, 135, 14, 20, 50, 57, 47, 69, 21, 55, 83, 115, 419, 87, 60, 210, 61, 42, 54, 26, 90, 28, 29, 35, 32, 63, 171, 52, 59, 138, 113, 180, 111, 48, 88, 39, 41, 621, 72, 22, 953, 230, 103, 207, 126, 64, 33, 243

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245822.
Other related permutations:  A091205, A245703, A245815.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

allocatemem(234567890);
v014580 = vector(2^18);
v091226 = vector(2^22);
v091242 = vector(2^22);
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, j++; v091242[j] = n; v091226[n] = v091226[n-1]); n++);
A014580(n) = v014580[n];
A091226(n) = v091226[n];
A091242(n) = v091242[n];
A091205(n) = if(n<=1, n, if(isA014580(n), prime(A091205(A091226(n))), {my(irfs, t); irfs=subst(lift(factor(Mod(1, 2)*Pol(binary(n)))), x, 2); irfs[,1]=apply(t->A091205(t), irfs[,1]); factorback(irfs)}));
A245703(n) = if(1==n, 1, if(isprime(n), A014580(A245703(primepi(n))), A091242(A245703(n-primepi(n)-1))));
A245821(n) = A091205(A245703(n));
for(n=1, 10001, write("b245821.txt", n, " ", A245821(n)));

(define (A245821 n) (A091205 (A245703 n)))

a(n) = A091205(A245703(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245815(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A179278 Largest nonprime integer <= n.

Original entry on oeis.org

1, 1, 1, 4, 4, 6, 6, 8, 9, 10, 10, 12, 12, 14, 15, 16, 16, 18, 18, 20, 21, 22, 22, 24, 25, 26, 27, 28, 28, 30, 30, 32, 33, 34, 35, 36, 36, 38, 39, 40, 40, 42, 42, 44, 45, 46, 46, 48, 49, 50, 51, 52, 52, 54, 55, 56, 57, 58, 58, 60, 60, 62, 63, 64, 65, 66, 66, 68, 69, 70, 70, 72

Offset: 1

Reinhard Zumkeller, Jul 08 2010

			From _Gus Wiseman_, Dec 04 2024: (Start)
The nonprime integers <= n:
  1  1  1  4  4  6  6  8  9  10  10  12  12  14  15  16
           1  1  4  4  6  8  9   9   10  10  12  14  15
                 1  1  4  6  8   8   9   9   10  12  14
                       1  4  6   6   8   8   9   10  12
                          1  4   4   6   6   8   9   10
                             1   1   4   4   6   8   9
                                     1   1   4   6   8
                                             1   4   6
                                                 1   4
                                                     1
(End)

Cf. A014684, A113523, A113638, A062298.
For prime we have A007917.
For nonprime we have A179278 (this).
For squarefree we have A070321.
For nonsquarefree we have A378033.
For prime power we have A031218.
For non prime power we have A378367.
For perfect power we have A081676.
For non perfect power we have A378363.
A000040 lists the primes, differences A001223.
A002808 lists the composite numbers, differences A073783.
A018252 lists the nonprimes, differences A065310.
A095195 has row n equal to the k-th differences of the prime numbers.
A113646 gives least nonprime >= n.
A151800 gives the least prime > n, weak version A007918.
A377033 has row n equal to the k-th differences of the composite numbers.

Array[# - Boole[PrimeQ@ #] - Boole[# == 3] &, 72] (* Michael De Vlieger, Oct 13 2018 *)
Table[Max@@Select[Range[n],!PrimeQ[#]&],{n,30}] (* Gus Wiseman, Dec 04 2024 *)

a(n) = if (isprime(n), if (n==3, 1, n-1), n); \\ Michel Marcus, Oct 13 2018

For n > 3: a(n) = A113523(n) = A014684(n);
For n > 0: a(n) = A113638(n). - Georg Fischer, Oct 12 2018
A005171(a(n)) = 1; A010051(a(n)) = 0.
a(n) = A018252(A062298(n)). - Ridouane Oudra, Aug 22 2025

Inequality in the name reversed by Gus Wiseman, Dec 05 2024

A245822 Permutation of natural numbers: a(n) = A245704(A091204(n)).

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 9, 6, 16, 11, 10, 19, 33, 12, 25, 17, 15, 23, 34, 39, 70, 13, 24, 26, 50, 21, 52, 53, 18, 31, 55, 77, 93, 54, 22, 29, 27, 66, 105, 67, 48, 137, 156, 30, 28, 37, 64, 91, 35, 85, 58, 97, 49, 40, 98, 36, 135, 59, 45, 47, 261, 56, 76, 92, 122, 83, 374, 38, 102, 139, 69, 167, 130, 88, 203, 351, 212, 349, 235, 14

Offset: 1

Antti Karttunen, Aug 02 2014

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10001
Index entries for sequences that are permutations of the natural numbers

Inverse: A245821.
Other related permutations: A091204, A245704, A245816.
Fixed points: A245823.
Cf. A000040, A049084, A078442, A049076, A018252, A062298.

allocatemem(123456789);
default(primelimit, 2^22)
v014580 = vector(2^18); A014580(n) = v014580[n];
v091226 = vector(2^22); A091226(n) = v091226[n];
A002808(n)={ my(k=-1); while( -n + n += -k + k=primepi(n), ); n}; \\ This function from M. F. Hasler
isA014580(n)=polisirreducible(Pol(binary(n))*Mod(1, 2)); \\ This function from Charles R Greathouse IV
i=0; j=0; n=2; while((n < 2^22), if(isA014580(n), i++; v014580[i] = n; v091226[n] = v091226[n-1]+1, v091226[n] = v091226[n-1]); n++)
A091204(n) = if(n<=1, n, if(isprime(n), A014580(A091204(primepi(n))), {my(pfs, t, bits, i); pfs=factor(n); pfs[,1]=apply(t->Pol(binary(A091204(t))), pfs[,1]); sum(i=1, #bits=Vec(factorback(pfs))%2, bits[i]<<(#bits-i))}));
A091245(n) = ((n-A091226(n))-1);
A245704(n) = if(1==n, 1, if(isA014580(n), prime(A245704(A091226(n))), A002808(A245704(A091245(n)))));
A245822(n) = A245704(A091204(n));
for(n=1, 10001, write("b245822.txt", n, " ", A245822(n)));

(define (A245822 n) (A245704 (A091204 n)))

a(n) = A245704(A091204(n)).
Other identities. For all n >= 1, the following holds:
A078442(a(n)) = A078442(n), A049076(a(n)) = A049076(n). [Preserves "the order of primeness of n"].
a(p_n) = p_{a(n)} where p_n is the n-th prime, A000040(n).
a(n) = A049084(a(A000040(n))). [Thus the same permutation is induced also when it is restricted to primes].
A245816(n) = A062298(a(A018252(n))). [While restriction to nonprimes induces another permutation].

A065864 Remainder when n is divided by the number of nonprimes not exceeding n.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 1, 0, 4, 4, 5, 5, 6, 6, 6, 6, 7, 7, 8, 8, 8, 8, 9, 9, 9, 9, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 14, 14, 14, 14, 15, 15, 15, 15, 15, 15, 16, 16, 16, 16, 16, 16, 17, 17, 18, 18, 18, 18, 18, 18, 19, 19, 19, 19, 20, 20, 21, 21, 21, 21, 21, 21

Offset: 1

Labos Elemer, Nov 26 2001

nonn

			For n=100, pi(100)=25, so a(100) = 100 mod (100-25) = 25.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000040, A004648, A062298, A065133, A065134, A065858-A065864.

Table[Mod[n, n - PrimePi@ n], {n, 78}] (* or *)
Table[Mod[n, Count[Range@ n, k_ /; ! PrimeQ@ k]], {n, 78}] (* Michael De Vlieger, Jan 01 2017 *)

{ for (n = 1, 1000, a=n%(n - primepi(n)); write("b065864.txt", n, " ", a) ) } \\ Harry J. Smith, Nov 02 2009

a(n) = n mod (n-pi(n)) = n mod (n-A000720(n)) = n mod A062298(n).

cp's OEIS Frontend

A254218 T(n,k) = number of length n 1..(k+2) arrays with no leading or trailing partial sum equal to a prime and no consecutive values equal.

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A254539 T(n,k)=Number of length n 1..(k+2) arrays with no leading partial sum equal to a prime.

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A255716 T(n,k)=Number of length n 1..(k+2) arrays with no leading partial sum equal to a prime and no consecutive values equal.

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A239968 a(n) = 0 unless n is a nonprime A018252(k) then a(n) = k.

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A356068 Number of integers ranging from 1 to n that are not prime-powers (1 is not a prime-power).

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Mathematica

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A065134 Remainder when n is divided by the number of primes not exceeding n.

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PARI

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A245821 Permutation of natural numbers: a(n) = A091205(A245703(n)).

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PARI

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A179278 Largest nonprime integer <= n.

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A245822 Permutation of natural numbers: a(n) = A245704(A091204(n)).

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